Key Equations

• Derivative of a vector-valued function
  [latex]{\bf{r}}^{\prime}(t)=\underset{\Delta{t}\to{0}}{\lim}\frac{{\bf{r}}(t+\Delta{t})-{\bf{r}}(t)}{\Delta{t}}[/latex]
• Principal unit tangent vector
  [latex]{\bf{T}}(t)=\dfrac{{\bf{r}}^{\prime}(t)}{\parallel{\bf{r}}^{\prime}(t)\parallel}[/latex]
• Indefinite integral of a vector-valued function
  [latex]\displaystyle\int [f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}]dt=\left[\displaystyle\int f(t)dt\right]{\bf{i}}+\left[\displaystyle\int g(t)dt\right]{\bf{j}}+\left[\displaystyle\int h(t)dt\right]{\bf{k}}[/latex]
• Definite integral of a vector-valued function
  [latex]\displaystyle\int_{a}^{b} [f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}]dt=\left[\displaystyle\int_{a}^{b} f(t)dt\right]{\bf{i}}+\left[\displaystyle\int_{a}^{b} g(t)dt\right]{\bf{j}}+\left[\displaystyle\int_{a}^{b} h(t)dt\right]{\bf{k}}[/latex]

Glossary

definite integral of a vector-valued function

the vector obtained by calculating the definite integral of each of the component functions of a given vector-valued function, then using the results as the components of the resulting function

derivative of a vector-valued function

the derivative of a vector-valued function [latex]{\bf{r}}(t)[/latex] is [latex]{\bf{r}}^{\prime}(t)=\underset{\Delta{t}\to{0}}{\lim}\frac{{\bf{r}}(t+\Delta{t})-{\bf{r}}(t)}{\Delta{t}}[/latex], provided the limit exists

indefinite integral of a vector-valued function

a vector-valued function with a derivative that is equal to a given vector-valued function

principal unit tangent vector

a unit vector tangent to a curve [latex]C[/latex]

tangent vector

to [latex]{\bf{r}}(t)[/latex] at [latex]t=t_{0}[/latex] any vector [latex]{\bf{v}}[/latex] such that, when the tail of the vector is placed at point [latex]{\bf{r}}(t_{0})[/latex] on the graph, vector [latex]{\bf{v}}[/latex] is tangent to curve [latex]C[/latex]